Quantum simulation of Burgers turbulence: Nonlinear transformation and direct evaluation of statistical quantities

TL;DR

This paper introduces a quantum algorithm for solving the nonlinear Burgers equation by leveraging the Cole-Hopf transformation, enabling efficient extraction of statistical properties and offering exponential speedup over classical methods.

Contribution

The paper presents a novel quantum algorithm that transforms the nonlinear Burgers equation into a linear form and efficiently extracts statistical quantities, demonstrating potential exponential computational advantages.

Findings

Quantum algorithm successfully solves Burgers equation using Cole-Hopf transformation.

Efficient extraction of multi-point functions from quantum states.

Exponential speedup over classical finite difference methods under certain conditions.

Abstract

Fault-tolerant quantum computing is a promising technology to solve linear partial differential equations that are classically demanding to integrate. It is still challenging to solve non-linear equations in fluid dynamics, such as the Burgers equation, using quantum computers. We propose a novel quantum algorithm to solve the Burgers equation. With the Cole-Hopf transformation that maps the fluid velocity field $u$ to a new field $\psi$, we apply a sequence of quantum gates to solve the resulting linear equation and obtain the quantum state $\vert\psi\rangle$ that encodes the solution $\psi$. We also propose an efficient way to extract stochastic properties of $u$, namely the multi-point functions of $u$, from the quantum state of $\vert\psi\rangle$. Our algorithm offers an exponential advantage over the classical finite difference method in terms of the number of spatial grids when a perturbativity condition in the information-extracting step is met.